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Adding a threshold to random ballot

One of the first things almost everyone who doesn’t instantly reject the idea suggests is that they’d be more comfortable with random ballot if it had a threshold to keep the really fringe people out – say anyone who would lose their deposit under the current UK system because they have fewer than 5% of the vote is disqualified and can’t elected. The idea being that although it’s a low-probability event, you’re going to still get a sizable minority of MPs elected with less than 5% of the vote (about \(\frac{1}{20}\) of them in fact, so in the UK around 32.5), which is pretty unfair to those constituents (in a way that apparently getting elected with only 25% of the vote is not).

My major problems with this proposal are:

It distorts the proportionality properties of random ballot

It disenfranchises anyone who votes for someone who failed to reach threshold – which may be a lot more than that fraction of the population if there are multiple minority candidates!

It takes a tactics free voting system and reintroduces something that looks like a pretty classic spoiler vote

How much I care about each of these points varies:

I don’t really have a problem with this as long as the threshold is low enough (and 5% is) – proportional representation is probably better when you restrict the candidates to the subset of the possible representatives that aren’t universally reviled.

This is bad and to be avoided to the greatest degree possible.

Eh, if I’m going to insist on tactics free voting my options are pretty limited. It would be nice to reduce the amount of tactical voting though.

I think I’ve figured out a patch which fixes the second and mitigates the third. The result is a curious hybrid system that is somewhere between random ballot, approval voting and instant runoff voting, but I actually think it works pretty well and, complexity aside, might be strictly better than random ballot for a lot of use cases.

The key idea is this: We make it a ranked system, where your ranked vote is interpreted to mean that you wish your vote to be cast for the highest ranked candidate on your ballot that has not been disqualified for failing to clear the threshold. Voters neither can nor should rank every candidate – the only candidates that appear on their ballots should be ones they’d be happy to vote for.

We then arrange a set of qualified candidates so that every candidate who qualifies has at least a threshold worth of votes cast for them under the above interpretation.

The system works as follows.

Once the ballots have been counted, the threshold is calculated as a number of ballots. This is now fixed and does not get updated throughout the process.

Each candidate is marked as either “uncertain”, “disqualified”, or “eligible”. Every candidate starts as uncertain. Each ballot is marked as either “available”, “withdrawn” or cast”. Every ballot starts as available.

A ballot approves a candidate if the candidate appears on the vote’s ballot. A ballot prefers a candidate if it approves them and every candidate it ranks more highly is disqualified.

Ballot counting now proceeds in a number of rounds until there are no more available votes. A round proceeds as follows:

Every uncertain candidate which is preferred by more than a threshold worth of ballots is marked as eligible.

Every available ballot whose preferred candidate is eligible is marked as cast. This includes candidates marked eligible in previous rounds.

Every uncertain candidate that is approved by fewer than a threshold worth of available ballots is marked as disqualified

If no uncertain candidate has changed its state this round (either by being marked eligible or disqualified), the candidate with the fewest approvals is marked as disqualified. In the event of a tie, the candidate that is preferred by fewer available votes drops out. If that is still a tie, break it arbitrarily.

Every available ballot whose approved candidates have all been disqualified is marked as withdrawn.

At the end, we have a set of eligible candidates and every vote is either cast or withdrawn after they could not collaborate with enough other people to get a candidate they liked elected. Random ballot is now performed with the cast ballots, with the preferred candidate of the selected ballot being elected.

Example

Let’s see how this might play out in practice. We’ll use an unrealistically high threshold of 40% so that we can see how it works properly without requiring a large number of candidates.

Pat does not have enough approvals to meet the threshold, so drops out.

The 4% Pat, Alex ballots are now cast for Alex who now has 44%

The 1% Pat voters are marked as withdrawn

Both Kim and Charlie have enough approvals to make the threshold, but neither meets the threshold, so the one with the lowest approvals (which is Kim at 30% vs Charlie’s 55%) drops out.

All remaining available votes prefer Charlie, so Charlie is marked as eligible with 55% of the vote.

So at the end, Alex has 44% of the vote, Charlie has 55%, and the remaining 1% of the votes are wasted. This means that Charlie wins with probability \(\frac{55}{99} = \frac{5}{9} \approx 56%\).

Reasoning

The core idea is to select a set of candidates that you can just use normal random ballot on and have everyone automatically clear the threshold. The election procedure is essentially a greedy algorithm to achieve that.

The motivation behind using approval voting for the selection stage is that immediately dropping out all candidates who aren’t approved by at least of threshold is always the correct thing to do: They can no longer get elected (because there aren’t enough votes that could be cast for them to clear the threshold), so having them stay in the game just leaves them as potential spoilers.

Once you have that, using approval voting to decide dropouts is also the obvious thing to do: The candidate that drops out is the one most likely to drop out later anyway. Approval voting is also a much more stable metric to use for deciding dropouts than preferences.

A simpler version of this system might be to simply mark everyone who gets at least a threshold worth of approvals as eligible. The major reasons I don’t like that is that you can get candidates through who have fewer than a threshold of votes cast for them (which isn’t awful but feels non-ideal) and that it brings in additional tactical elements about whether or not to approve candidates (because if you don’t their votes might go to you).

Evaluation

(Note: This is an informal evaluation. I haven’t actually done simulations to verify this yet)

I think this works pretty well. It mostly works out like random ballot with quality control on which candidates actually run.

The only people who are disenfranchised entirely by this system are the people who so hate any candidate that at least the critical threshold of the voting population can tolerate that they can’t bring themselves to vote for that candidate. Note that this is a much smaller set of people than those who merely prefer every such candidate to any mainstream one. I… probably should still have a problem with disenfranchising these people, but I can’t really bring myself to right now. So I’m calling that a win for point 2.

So what does tactical voting look like under this system?

I think it’s pretty mild.

First off, there is never any point in ranking candidates in an order other than your true preferences. The only time that your rank order matters between two candidates is when both of them are eligible, so the usual argument for random ballot holds. So the only question of tactics is who to include on your ballot.

If you’re comfortable that your favourite candidate will be eligible, you can stop there. In general as soon as you’ve ranked a candidate who you’re sure will make threshold there’s no point in going further.

In particular this means that everyone who would under random ballot cast their vote for a candidate who makes it past the threshold of first choices doesn’t need to worry about the rest of the system and can just comfortably cast a single vote as per normal. If every candidate clears the threshold then this system then just reduces to random ballot.

Tactics then consist of essentially jockeying a bit to try to get minority candidates you like who haven’t quite cleared the threshold to get enough votes so that they can be eligible.

There’s no point in omitting a candidate from your ballot when you’re certain they’re going to make threshold on first choice votes before anything else of import has happened. Once a candidate has been made eligible their presence or absence on your ballot has no impact one way or the other. So generally tactical voters will pick a backstop candidate who they’re sure is threshold clearing as their preferred mainstream option if all else fails.

I think the same mostly applies for anyone you’re morally certain is going to clear the threshold before anyone you prefer to them. This isn’t perfectly true – it may be in your interests to keep them in the running for longer so you get to keep counting their approvals for candidates you like to stop said candidates dropping out. This is a sufficiently hard to predict move that I don’t expect it to have much practical impact.

So the only real question of tactics is which other fringe candidates you approve. Basically what you want to do is ensure that your lower ranked preferences don’t end up competing with your higher ranked preferences for who gets disqualified.

So, in other words, the easiest tactical vote consists of casting your ballot as follows:

Always rank your favourite candidate first.

Repeatedly add your favourite candidate of the candidates who are “significantly more mainstream” than the ones you’ve added so far.

Keep adding candidates as per above until you’ve added one you are certain will meet threshold.

I would be surprised if many voters bothered to rank more than three candidates – their actual favourite, their second choice who might make threshold, and the mainstreamish candidate (for values of mainstream that mean “have the support of at least 5% of the population”) who they like most.

So essentially the tactical voting creates a bias towards voting for making your subsequent preferences a bit more mainstream than you otherwise would.

But that’s what the system is intended to do anyway. So if that’s genuinely all the tactical bias there is, if anything it’s a feature.

Assuming I’ve not missed some critical flaw under tactical voting, all told I think this is a pretty neat system which nicely balances the proportionality properties of random ballot with a mild mainstream bias.

Applications

This should be applicable most places where random ballot is, assuming the complexity cost is not prohibitive (e.g. you probably don’t want to use this for deciding where to go for lunch).

Where the additional complexity isn’t a problem (e.g. because vote tallying is computerised anyway) I think I might actually just flat out prefer this system to random ballot – it’s a good way to curb some of its variance while still retaining most of its desirable properties.

This is particularly true for cases where random ballot is not currently entirely suitable (because e.g. the stakes are very high and/or the number of repetitions of the vote are quite low) – if you were to push the threshold up to somewhere more in the region of 40% it might even become suitable for more presidential style single winner elections (or, at least, not obviously less suitable than the current systems).

It might also be an interesting procedure for other allocation problems. There’s nothing intrinsic to random ballot for most of it – it’s just a question of how to divide up numbers so that they add up to 100% (in the event that nobody withdraws their vote) and each meet a threshold. You could just as easily use it for e.g. deciding how to distribute money between a number of projects or charities.